Recently, the study of near- and below- threshold regime harmonics as a potential source of intense coherent vacuum-ultraviolet radiation has received considerable attention. However, the dynamical origin of these lower harmonics, particularly for the molecular systems, is less understood and largely unexplored. Here we perform the first fully ab initio and high precision 3D quantum study of the below- and near-threshold harmonic generation of molecules in an intense 800-nm near-infrared (NIR) laser field. Combining with a synchrosqueezing transform of the quantum time-frequency spectrum and an extended semiclassical analysis, we explore in-depth the roles of various quantum trajectories, including short- and long trajectories, multiphoton trajectories, resonance-enhanced trajectories, and multiple rescattering trajectories of the below- and near- threshold harmonic generation processes. Our results shed new light on the dynamicalorigin of the below- and near-threshold harmonic generation and various quantum trajectories for diatomic molecules for the first time.

f3: Semiclassical trajectories: Energy, position, and time, and scheme of electron dynamics.(a) Semiclassical return energy as a function of several ionization times and return times. The solid and dotted colored lines represent the short and long trajectories, respectively. Here we can see the multi-rescattering of the long trajectories at different return times. (b) For clarity, we show the semiclassical return energy as a function of ionization time (black line) and return time (green line) for the electrons released during one optical cycle preceding the pulse peak. Several typical rescatterings are marked by 1’ (blue text) short trajectory, 1 (black text) long trajectory (E > 0) and 1 (red text) long trajectory (E = 0), and multi-rescattering trajectories are marked by 2–5 (red text). (c) Position vs time in below- and near-threshold regions for the corresponding return energies shown in (b). The black horizontal dashed lineindicates the position of the two hydrogen nuclei (z = ±1 a.u.) for and the colored dots are to help guide the eye for the return times for different trajectories. Here the initial condition is that the electrons with an initial velocity v0 are released from the left side hydrogen core (z = −1 a.u.) along the electronic-field force Fz. The green solid line indicates the corresponding laser field, and the laser parameters used are the same as those in Fig. 1.

Mentions:
The time-frequency representation in Fig. 2 shows a periodic repetition of arches comprising the short and long trajectories. It is readily observed that the main contribution to the above-threshold harmonics is due to the short trajectories (dense red color in Fig. 2). The prominent trajectory located near the vicinity of the 7th harmonic is the 1σg–1σu multiphoton resonance-transition of . To explore the dynamical role of the quantum trajectories, we extend a standard semiclassical approach suggested independently by Corkum11 and Kulander et al.12 with the inclusion of the molecular potential. Here the electric-field force corresponding to the applied laser field in atomic units isFz = E(t)ez, where ez is the unit vector in the z direction and E(t) is the electric field of the laser pulse. For the laser parameters used, the corresponding Keldysh parameter γ is >1 and the multiphoton ionization regime is expected to be dominant, the initial conditions are provided by releasing the electrons with an initial velocity (v0) to overcome a potential barrier. Therefore, the direction of the initial velocity of electrons is either ‘identical’ or ‘opposite’ with respect to Fz. Note that when the direction of the initial velocity of electrons is ‘identical’, the electron gains an extra energy to escape the barrier; when the direction of the initial velocity of electrons is‘opposite’, the electron is pushed back when leaving the barrier. The semiclassical return energy as a function of the ionization time and return time of the electrons that are released in the first one cycle before the pulse peak for ‘identical’ conditions are presented in Fig. 2 overlaying the SST time-frequency analysis. We indicate the short trajectories (green solid line) and long trajectories (green dashed line) as those in the standard three-step model, as well as the multi-rescattering trajectories [green dashed line (T ~ 0.8 − 1.75)]. In Fig. 2, by comparing with the classical calculation, it is clearly seen that the multirescattering trajectory (green dashed line; T > 0.7 optical cycles) has strong contributions to the below- and near-thresholdharmonic 18–27. Of course, harmonics 18 to 27 have contributions from the short trajectories during subsequent half-cycles. However, it is clearly seen that the multirescattering trajectory has its strong contributions as well. This result is in good agreement with the semiclassical result shown in Supplementary Fig. 1. Particularly, several multirescattering trajectories are superposed at around 0.7–1.75 optical cycles, which contribute to the generation of the below- and near-threshold harmonics. To explore the intricate structures in the below-, near-, and above-threshold generation, we show the semiclassical return energy as a function of several ionization times and return times in Fig. 3a and b. Note that the returning energy includes the kinetic energy and the potential energy, and thus may become negative below the ionization threshold. It is clearly seen that the HHGoriginated from two quantum trajectories, namely, the short trajectories (the higher harmonics are emitted after the lower ones) and the long trajectories (the higher harmonics are emitted before the lower ones). In Fig. 3a and b, the trajectories that lie between the harmonic 15 and 37 suggest multiple returns of the electron. In Fig. 3b, for clarity, we show the semiclassical return energy as a function of ionization time (black line) and return time (green line) for the electrons released during one optical cycle preceding the pulse peak. We label the short trajectories 1′ and long trajectories 1, as well as the multi-rescattering trajectories 2–5. We find that the long-trajectories electrons associated with the multirescatterings (see the black dashed arrow) contribute to the below- and near-threshold harmonics 17–27, and the corresponding travel time is larger than one opticalcycle. The trajectories that are released late and return early are regarded as the short trajectories 1′, while those released early and return late are the long trajectories 1–5. The trajectories 1′ and 1–5 are superimposed in Fig. 2 for the sake of comparison with the SST representation. As shown in the figures, the structures of the SST representation are in agreement with the semiclassical trajectories, although the semiclassical result for the long trajectories are a little broader in time for the first and second return. This can be seen in the range from T ~ 0.4 − 0.6 (horizontal yellow arrow), where the semiclassical first return long trajectory (green dashed line) is a little broader than the SST time-frequency analysis. Also, the semiclassical second return multirescattering trajectory (green dashed line) inthe range from T ~ 0.8 − 1.0 (horizontal purple arrow) is broader in time than seen in the SST representation. Trajectories with ‘identical’ (our initial conditions) and ‘opposite’ initial conditions interchange because the laser field changes sign every half optical cycle. In the quantum mechanical SST representation, trajectories with both ‘identical’ and ‘opposite’ initial conditions all appear in one optical cycle.

f3: Semiclassical trajectories: Energy, position, and time, and scheme of electron dynamics.(a) Semiclassical return energy as a function of several ionization times and return times. The solid and dotted colored lines represent the short and long trajectories, respectively. Here we can see the multi-rescattering of the long trajectories at different return times. (b) For clarity, we show the semiclassical return energy as a function of ionization time (black line) and return time (green line) for the electrons released during one optical cycle preceding the pulse peak. Several typical rescatterings are marked by 1’ (blue text) short trajectory, 1 (black text) long trajectory (E > 0) and 1 (red text) long trajectory (E = 0), and multi-rescattering trajectories are marked by 2–5 (red text). (c) Position vs time in below- and near-threshold regions for the corresponding return energies shown in (b). The black horizontal dashed lineindicates the position of the two hydrogen nuclei (z = ±1 a.u.) for and the colored dots are to help guide the eye for the return times for different trajectories. Here the initial condition is that the electrons with an initial velocity v0 are released from the left side hydrogen core (z = −1 a.u.) along the electronic-field force Fz. The green solid line indicates the corresponding laser field, and the laser parameters used are the same as those in Fig. 1.

Mentions:
The time-frequency representation in Fig. 2 shows a periodic repetition of arches comprising the short and long trajectories. It is readily observed that the main contribution to the above-threshold harmonics is due to the short trajectories (dense red color in Fig. 2). The prominent trajectory located near the vicinity of the 7th harmonic is the 1σg–1σu multiphoton resonance-transition of . To explore the dynamical role of the quantum trajectories, we extend a standard semiclassical approach suggested independently by Corkum11 and Kulander et al.12 with the inclusion of the molecular potential. Here the electric-field force corresponding to the applied laser field in atomic units isFz = E(t)ez, where ez is the unit vector in the z direction and E(t) is the electric field of the laser pulse. For the laser parameters used, the corresponding Keldysh parameter γ is >1 and the multiphoton ionization regime is expected to be dominant, the initial conditions are provided by releasing the electrons with an initial velocity (v0) to overcome a potential barrier. Therefore, the direction of the initial velocity of electrons is either ‘identical’ or ‘opposite’ with respect to Fz. Note that when the direction of the initial velocity of electrons is ‘identical’, the electron gains an extra energy to escape the barrier; when the direction of the initial velocity of electrons is‘opposite’, the electron is pushed back when leaving the barrier. The semiclassical return energy as a function of the ionization time and return time of the electrons that are released in the first one cycle before the pulse peak for ‘identical’ conditions are presented in Fig. 2 overlaying the SST time-frequency analysis. We indicate the short trajectories (green solid line) and long trajectories (green dashed line) as those in the standard three-step model, as well as the multi-rescattering trajectories [green dashed line (T ~ 0.8 − 1.75)]. In Fig. 2, by comparing with the classical calculation, it is clearly seen that the multirescattering trajectory (green dashed line; T > 0.7 optical cycles) has strong contributions to the below- and near-thresholdharmonic 18–27. Of course, harmonics 18 to 27 have contributions from the short trajectories during subsequent half-cycles. However, it is clearly seen that the multirescattering trajectory has its strong contributions as well. This result is in good agreement with the semiclassical result shown in Supplementary Fig. 1. Particularly, several multirescattering trajectories are superposed at around 0.7–1.75 optical cycles, which contribute to the generation of the below- and near-threshold harmonics. To explore the intricate structures in the below-, near-, and above-threshold generation, we show the semiclassical return energy as a function of several ionization times and return times in Fig. 3a and b. Note that the returning energy includes the kinetic energy and the potential energy, and thus may become negative below the ionization threshold. It is clearly seen that the HHGoriginated from two quantum trajectories, namely, the short trajectories (the higher harmonics are emitted after the lower ones) and the long trajectories (the higher harmonics are emitted before the lower ones). In Fig. 3a and b, the trajectories that lie between the harmonic 15 and 37 suggest multiple returns of the electron. In Fig. 3b, for clarity, we show the semiclassical return energy as a function of ionization time (black line) and return time (green line) for the electrons released during one optical cycle preceding the pulse peak. We label the short trajectories 1′ and long trajectories 1, as well as the multi-rescattering trajectories 2–5. We find that the long-trajectories electrons associated with the multirescatterings (see the black dashed arrow) contribute to the below- and near-threshold harmonics 17–27, and the corresponding travel time is larger than one opticalcycle. The trajectories that are released late and return early are regarded as the short trajectories 1′, while those released early and return late are the long trajectories 1–5. The trajectories 1′ and 1–5 are superimposed in Fig. 2 for the sake of comparison with the SST representation. As shown in the figures, the structures of the SST representation are in agreement with the semiclassical trajectories, although the semiclassical result for the long trajectories are a little broader in time for the first and second return. This can be seen in the range from T ~ 0.4 − 0.6 (horizontal yellow arrow), where the semiclassical first return long trajectory (green dashed line) is a little broader than the SST time-frequency analysis. Also, the semiclassical second return multirescattering trajectory (green dashed line) inthe range from T ~ 0.8 − 1.0 (horizontal purple arrow) is broader in time than seen in the SST representation. Trajectories with ‘identical’ (our initial conditions) and ‘opposite’ initial conditions interchange because the laser field changes sign every half optical cycle. In the quantum mechanical SST representation, trajectories with both ‘identical’ and ‘opposite’ initial conditions all appear in one optical cycle.

Recently, the study of near- and below- threshold regime harmonics as a potential source of intense coherent vacuum-ultraviolet radiation has received considerable attention. However, the dynamical origin of these lower harmonics, particularly for the molecular systems, is less understood and largely unexplored. Here we perform the first fully ab initio and high precision 3D quantum study of the below- and near-threshold harmonic generation of molecules in an intense 800-nm near-infrared (NIR) laser field. Combining with a synchrosqueezing transform of the quantum time-frequency spectrum and an extended semiclassical analysis, we explore in-depth the roles of various quantum trajectories, including short- and long trajectories, multiphoton trajectories, resonance-enhanced trajectories, and multiple rescattering trajectories of the below- and near- threshold harmonic generation processes. Our results shed new light on the dynamicalorigin of the below- and near-threshold harmonic generation and various quantum trajectories for diatomic molecules for the first time.